Plus ça change, plus c’est la même chose [The more that changes, the more it is the same thing.] - Jean Baptiste Alphonse Karr, 1849
What are the time and frequency domains? A filter’s action can be described by its impulse response as well as by its frequency spectrum. The filter’s impulse response is in the time domain, as is the input signal itself. The filter’s frequency spectrum is in the frequency domain. Both modes of expression are functions of each other - that is, if one is known, the other can be derived from it.
In digital filtering, either domain can be employed, but generally, both the seismic signal and the characteristics of the filter must be converted into the same form. For example, we need for the operation to be in the time domain, but only the frequency spectrum of the filter is specified. In such a case, the frequency spectrum must be transformed into an impulse response (in the time domain) so that operation can be carried out in the time domain. The Fourier transform and the inverse Fourier transform provide the physical basis for such conversions from one domain to the other.
What is a Fourier transform? The Fourier transform converts a function of time (the signal) into the corresponding function of frequency (the temporal frequency spectrum). The inverse Fourier transform works in the reverse direction. Specifically, the inverse Fourier transform converts a function of frequency (the temporal frequency spectrum) into the corresponding function of time (the signal).
The Fourier transform need not apply only to frequency in cycles per second and time in seconds. The Fourier transform also can be used for spatial measurements. For example, consider the case of a coherent spatial distribution across a receiver aperture. In such a case, a relationship exists between the transmission pattern of the system in terms of the sine of the angle of projection and the distribution of the field along the aperture of the system (Robinson, 1967). The temporal frequency (i.e., the number of cycles per unit time) is the Fourier dual for the time (i.e., the number of time units). In the same way, the spatial frequency, or wavenumber, is the Fourier dual for the space variable. The wavenumber gives the number of cycles per unit distance. In this chapter, we deal only with time and temporal frequency, so we shorten the expression temporal frequency spectrum to frequency spectrum.
Fourier transform theory often is applicable even when the variables involved may appear to have no direct physical meaning. For example, with use of the Fourier transform, the convolution of two time functions can be expressed in terms of the multiplication of their associated Fourier transforms.
Spectral analysis plays an important role in geophysics, as it does in all science. See, for example, Robinson (1970, 1982, 1983), Ulrych and Jensen (1974), Treitel et al. (1977), Gutowski et al. (1977, 1978), Treitel and Robinson (1981), Haykin et al. (1983), Rosa and Ulrych (1991), and Sacchi and Ulrych (1996). At this point, it is fitting to quote the inimitable words of Ulrych (2008, p. 1):
The object of seismic exploration is encoded in the data that are acquired on or near the surface of the earth. The goal of decoding these data is, essentially, to find out where and what this object is. Although we record our information in space and time, we always, at some stage, follow the teachings of Jean Baptiste Joseph Fourier and transform our measurements into the frequency domain. In this domain, our data live in the phase and temporal and spatial frequency dimensions. The “where” is encoded in the phase, the “what” is encoded in both the phase and amplitude.
- Robinson, E. A., 1967, Statistical communication and detection: Hafner Publishing Co.
- Robinson, E. A., 1970, Spectral model of geological time measurements: Proceedings of the IEEE 9th Symposium on Adaptive Processes, Decision and Control, University of Texas at Austin, 20.1.1–20.1.3.
- Robinson, E. A., 1982, Spectral approach to geophysical inversion by Lorentz, Fourier, and radon transforms: Proceedings of the IEEE, 70, 1039–1054.
- Robinson, E. A., 1983, Iterative least-squares procedure for ARMA spectral estimation, in S. Haykin, ed., Nonlinear methods of spectral analysis, 2nd ed.: Topics in Applied Physics, no. 34, Springer, 127–153.
- Ulrych, T., and O. Jensen, 1974, Cross-spectral analysis using maximum entropy: Geophysics, 39, 353–356.
- Treitel, S., E. A. Robinson, and P. R. Gutowski, 1977, Empirical spectral analysis revisited: in J. J. H. Miller, ed., Topics in numerical analysis, 3: Academic Press, 429–446.
- Gutowski, P. R., E. A. Robinson, and S. Treitel, 1977, Novel aspects of spectral estimation: Proceedings of the 1977 Joint Automatic Control Conference, 1, 99–104.
- Gutowski, P. R., E. A. Robinson, and S. Treitel, 1978, Spectral estimation, fact or fiction: IEEE Transactions on Geoscience Electronics, GE-16, 80–84.
- Treitel, S., and E. A. Robinson, 1981, Maximum entropy spectral decomposition of a seismogram into its minimum entropy component plus noise: Geophysics, 46, 1108–1115.
- Haykin, S., S. Kesler, and E. A. Robinson, 1983, Recent advances in spectral estimation, in S. Haykin, ed., Nonlinear methods of spectral analysis, 2nd ed.: Topics in Applied Physics, no. 34, Springer, 245–260.
- Rosa, A. L. R., and T. J. Ulrych, 1991, Processing via spectral modeling: Geophysics, 56, 1244–1251.
- Sacchi, M. D., and T. J. Ulrych, 1996, Estimation of the discrete Fourier transform, a linear inversion approach: Geophysics, 61, 1128–1136.
- Ulrych, T., 2008, The role of amplitude and phase in processing and inversion: SEG Distinguished Lecture Program, 2008, <http://ce.seg.org/dl/spring2008/index.shtml> accessed 22 March 2008.
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Also in this chapter
- Magnitude spectrum and phase spectrum
- Fourier transform
- Minimum-phase spectrum
- Inverse Fourier transform
- Appendix F: Exercises